Cantilevered airfoils, particularly those with high aspect ratios, are generally soft in torsion. Because the chordwise position of the center-of-fit is dependent on pitch angle, a lack of torsional stiffness can lead to oscillations in bending arising from non-uniform aerodynamic loading encountered during ostensibly normal flight maneuvers, both for fixed-wing and rotary-wing aircraft. The magnitude of such instabilities depend both on aeroelastic and aerodynamic factors, and can range from the imperceivable to the destructive, and are further accentuated if significant tip-weight is present, such as arising from armament, fuel tanks, or lift engines.
A common form of such instabilities is flutter which primarily manifests itself as wing tip or rotor tip oscillations in bending and torsion. In terms of aeroelastic factors, the amplitude of such bending excursions is inversely proportional to airfoil stiffness in torsion and flexure. In terms of aerodynamic factors, the amplitude of such bending excursions is dependent on airfoil section and thickness ratio, pitch angle and sweep angle. Moreover, the amplitude increases with increasing airspeed, becoming pronounced at airspeeds approaching sonic.
If the bending excursion results in structural twist that tends to increase the magnitude of the excursion, such as an upwards bending resulting in an increase in tip pitch angle, then the effect is destabilizing in regard to flutter oscillations, whether the aerodynamic loading is uniform or non-uniform. It follows therefore that an upwards bending accompanied of a decreasing tip pitch angle would lead to static stability, and would probably ameliorate dynamic stability, depending on the frequency and amplitude of the oscillations. If an airfoil bending excursion through section angle d.DELTA. results in a change in airfoil twisting through section angle d.alpha. then the airfoil exhibits torsional-flexural coupling; d.alpha./d.DELTA..noteq.0. Accordingly, if d.alpha./d.DELTA.&gt;0 the airfoil is statically unstable. Only if d.alpha./d.DELTA.&lt;0 is the airfoil stable. The structural consideration that influence this coupling must now be considered.
In general, conventional airfoil construction can be considered isotropic, with the flexure moduli essentially identical in all directions. Consequently the axis about which preferential bending occurs is normal to the shortest dimension (thickness) and in a direction that intersects the shortest chordwise extent of the airfoil, and will be denoted the normal bending axis.
As is illustrated in FIGS. 1A, 1B and 1C for conventional airfoil construction, the relationship between the axis of bending B and the resulting change in the pitch angle depends largely on the angle between the relative wind and the bending axis of the airfoil, which will be denoted .mu.. When the axis of bending B is the normal axis B.sub.o then .mu.=.mu..sub.o. All angles are positive when measured clockwise from the relative-wind direction. From geometric considerations EQU d.alpha./d.DELTA.=-sin(.mu.) (1).
The orientation of the normal bending axis .mu..degree. can now be directly related to the airfoil sweep angle .beta. by the relationship EQU .mu..sub.o =.beta.-90.degree.. (2)
For rearward swept wings .beta.&gt;90.degree. and therefore d.alpha./d.DELTA.&lt;0. Accordingly upward bending through an angle .DELTA. results in downward pitching through an angle .alpha.. Thus rearward sweep is stabilizing.
For non-swept wing .beta.=90.degree. and therefore d.alpha./d.DELTA.=0. Nevertheless, particularly for high-aspect ratio non-symmetrical airfoils bending can result in pitch instability inasmuch as center of lift shifts with angle of attack.
For forward swept wings .beta.&lt;90.degree. and therefore d.alpha./d.DELTA..ltoreq.0. Accordingly upward bending through an angle .DELTA. results in upward pitching through an angle .alpha.. Thus forward sweep is destabilizing. Moreover, aircraft supported by such wings are longitudinally unstable (comparable to an arrow flying backwards). However, there are distinct advantages to forward sweep. Such aircraft are extremely maneuverable, a distinct advantage for high-speed military aircraft.
Despite d.alpha./d.DELTA.&gt;0 electronic stability augmentation systems can permit stable flight by minutely adjusting control surfaces to dampen longitudinal excursions to the point where such excursions are essentially imperceptible. Nevertheless, bending instabilities can greatly complicate artificial stabilization, as such instabilities tends to adjust the entire airfoil to amplify such excursions, counteracting the effect of stability augmentation. To minimize such excursions swept-forward wings are constructed sufficiently stiff to hold such excursions to an acceptable level. The penalty of this brute-strength approach is excessive wing weight. Accordingly, the stability of an essentially isotropic airfoil for which .mu.=.mu..sub.o is directly related to its sweep angle.
Rotor blades, having extreme aspect ratios, and being subject to periodically varying air loads, can be particularly susceptible to oscillations, as shown in FIG. 1D. Such blades operate in a highly non-uniform air-flow field. Not only must rotor blades operate through an extreme range of angles-of-attack in alternately passing through the advancing and retreating portions of the rotor disc, but each flies in the wake of the preceding rotor blade. Flutter amplitudes can reach unacceptable levels as advancing blade tips approach sonic speed. As in the case of swept-forward wings, the present procedure to minimize such instabilities depend not only on tip design but on the brute-strength approach: blades sufficiently stiff to hold flutter amplitudes to acceptable levels. In addition wind gusts can amplify the effect of the instabilities discussed. Because .beta.=90.degree. and therefore d.alpha./d.DELTA.=0 such oscillations are not readily diminished.
Consider now an alternative approach. It would be possible to diminished bending instability if the bending axis could be skewed so that .mu.&gt;.mu..sub.o, introducing beneficial torsional-flexural coupling in an airfoil despite the sweep angle. Accordingly, from equation (2) only the condition EQU .mu.&gt;.beta.-90.degree. (3)
would have to be met. That is, preferential bending would occur not about the normal bending axis .mu.=.mu..sub.o as shown in FIGS. 1A, 1B, 1C and 1D, but rather about a skewed axis .mu..noteq..mu..sub.o of greater length as shown in FIGS. 2A, 2B, 2C and 2D for a stable airfoil, although both bending axes are still axes that are normal to the shortest dimension of the structure. With .mu.&gt;.mu..sub.o the coupling d.alpha./d.DELTA. can be made independent of sweep angle .beta..
To construct an airfoil whose bending axis is skewed from its normal bending axis requires a construction in which the flexural modulus of the spar (longitudinal load-bearing member) of the airfoil can be controlled in different directions relative to the longitudinal axis of the spar. This requirement can ostensibly be met using filament reinforced composite construction. Either essentially unidirectional resin-preimpregnated filaments (prepreg) or a fabric with preferred filament-orientation that is resin-impregnated after lay-up is wound about a non-structural core. A unidirectional filament layer is denoted a ply. Adjacent plies in the same direction are denoted parallel plies and adjacent plies in different directions are denoted cross plies. Adjacent plies 90.degree. to one another are denoted orthogonal plies.
Upon curing of the composite the resin bonds the plies into a unitary spar in which the orientation of the flexure axis greatly depends on the preferred angular orientation of the filaments relative to the longitudinal axis of the spar. This composite spar comprises the load-bearing member of the airfoil.
Because the tensile modulus of filament-resin composite construction in the direction parallel to the filament is perhaps 25 times greater than that in the transverse direction, composite structures can be highly anisotropic, depending on the stacking sequence of the plies, using roughly the rule of mixtures. This possible effect is illustrated in FIGS. 3A, using a filament laid down about a conventional core as an example.
Ostensibly the orientation of the ply laid over the upper surface of the core at helical angle .theta..sub.U to the longitudinal axis of the core should result in a preferred bending axis B.sub.U skewed from the normal bending axis B.sub.o. However, the ply laid over the lower surface of the core is oriented at the opposite helical angle of .theta..sub.L from the longitudinal axis, which will result in a preferred bending axis B.sub.L oppositely skewed from the normal bending axis B.sub.o. consequently, because the filaments are conventionally laid .theta..sub.U +.theta..sub.L =0, and therefore the bending axes B.sub.U and B.sub.L cross. Accordingly, the effect of skewing the bending axes is lost. As a result the skewed bending axes shown in FIGS. 2A, 2B, 2C and 2D cannot be realized with conventional composite construction using multiple parallel plies. Multiple orthogonal plies will not alter this situation as such ply would simply alternate the bending axes B.sub.U and B.sub.L.
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